Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,15)(12,13)(14,16)(17,18)(19,28)(20,22)(21,31)(23,27)(24,32)(25,30)(26,29);(1,3,5,8)(2,4,6,7)(10,13)(14,15)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,26)(27,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(1,2)(3,7)(4,8)(5,6)(9,11,12,16)(10,14,13,15)(17,20,23,30)(18,25,27,22)(19,29,32,21)(24,26,28,31);(17,21,23,29)(18,26,27,31)(19,30,32,20)(22,24,25,28);(1,4,5,7)(2,3,6,8)(17,22,23,25)(18,20,27,30)(19,31,32,26)(21,24,29,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366317);